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15 include "basic_2/substitution/llor.ma".
16 include "basic_2/substitution/lleq_alt.ma".
18 (* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
20 (* Properties on poinwise union for local environments **********************)
22 lemma llor_lleq_O: ∀R,L1,L2,T. llpx_sn R 0 T L1 L2 →
23 ∀L. L1 ⩖[T] L2 ≡ L → L2 ≡[T, 0] L.
24 #R #L1 #L2 #T #H1 #L #H2
25 elim (llpx_sn_llpx_sn_alt … H1) -H1 #HL12 #IH1
26 elim H2 -H2 #_ #HL1 #IH2
27 @lleq_intro_alt // #I2 #I #K2 #K #V2 #V #i #Hi #HnT #HLK2 #HLK
28 lapply (ldrop_fwd_length_lt2 … HLK) #HiL
29 elim (ldrop_O1_lt (Ⓕ) L1 i) // -HiL #I1 #K1 #V1 #HLK1
30 elim (IH1 … HLK1 HLK2) // -IH1
32 elim (IH2 … HLK1 HLK2 HLK) * /2 width=1 by conj/